• transformation
• initialization files
• default initialization
• Programs that incorporate this model:
  • align_warp

Transformation

Given a coordinate (x,y,z) in the standard file, the coordinates of the corresponding voxel in the reslice file (x',y',z') are given by the equations:

First order polynomial (4 coefficients per coordinate)

• x'=k_x1+k_x2x+k_x3y+k_x4z
• y'=k_y1+k_y2x+k_y3y+k_y4z
• z'=k_z1+k_z2x+k_z3y+k_z4z

Second order polynomial (10 coefficients per coordinate)

• x'=k_x1+k_x2x+k_x3y+k_x4z +k_x5x²+k_x6xy+k_x7y² +k_x8xz+k_x9yz+k_x10z²
• y'=k_y1+k_y2x+k_y3y+k_y4z +k_y5x²+k_y6xy+k_y7y² +k_y8xz+k_y9yz+k_y10z²
• z'=k_z1+k_z2x+k_z3y+k_z4z +k_z5x²+k_z6xy+k_z7y² +k_z8xz+k_z9yz+k_z10z²

Third order polynomial (20 coefficients per coordinate)

• Equations can be extrapolated from the first and second order polynomials and the first 20 terms from the ordered list of polynomial terms below

Fourth order polynomial (35 coefficients per coordinate)

• Equations can be extrapolated from the first and second order polynomials and the first 35 terms from the ordered list of polynomial terms below

Fifth order polynomial (56 coefficients per coordinate)

• Equations can be extrapolated from the first and second order polynomials and the first 56 terms from the ordered list of polynomial terms below

Sixth order polynomial (84 coefficients per coordinate)

• Equations can be extrapolated from the first and second order polynomials and the first 84 terms from the ordered list of polynomial terms below

Seventh order polynomial (120 coefficients per coordinate)

• Equations can be extrapolated from the first and second order polynomials and the first 120 terms from the ordered list of polynomial terms below

Eighth order polynomial (165 coefficients per coordinate)

• Equations can be extrapolated from the first and second order polynomials and the first 165 terms from the ordered list of polynomial terms below

Ninth order polynomial (220 coefficients per coordinate)

• Equations can be extrapolated from the first and second order polynomials and the first 220 terms from the ordered list of polynomial terms below

Tenth order polynomial (286 coefficients per coordinate)

• Equations can be extrapolated from the first and second order polynomials and the first 286 terms from the ordered list of polynomial terms below

Eleventh order polynomial (364 coefficients per coordinate)

• Equations can be extrapolated from the first and second order polynomials and the first 364 terms from the ordered list of polynomial terms below

Twelfth order polynomial (455 coefficients per coordinate)

• Equations can be extrapolated from the first and second order polynomials and the first 455 terms from the ordered list of polynomial terms below

Ordered List of Polynomial Terms

1. 1
2. x
3. y
4. z (end of 1st order)
5. x²
6. xy
7. y²
8. xz
9. yz
10. z² (end of 2nd order)
11. x³
12. x²y
13. xy²
14. y³
15. x²z
16. xyz
17. y²z
18. xz²
19. yz²
20. z³ (end of 3rd order)
21. x⁴
22. x³y
23. x²y²
24. xy³
25. y⁴
26. x³z
27. x²yz
28. xy²z
29. y³z
30. x²z²
31. xyz²
32. y²z²
33. xz³
34. yz³
35. z⁴ (end of 4th order)
36. x⁵
37. x⁴y
38. x³y²
39. x²y³
40. xy⁴
41. y⁵
42. x⁴z
43. x³yz
44. x²y²z
45. xy³z
46. y⁴z
47. x³z²
48. x²yz²
49. xy²z²
50. y³z²
51. x²z³
52. xyz³
53. y²z³
54. xz⁴
55. yz⁴
56. z⁵ (end of 5th order)
57. x⁶
58. x⁵y
59. x⁴y²
60. x³y³
61. x²y⁴
62. xy⁵
63. y⁶
64. x⁵z
65. x⁴yz
66. x³y²z
67. x²y³z
68. xy⁴z
69. y⁵z
70. x⁴z²
71. x³yz²
72. x²y²z²
73. xy³z²
74. y⁴z²
75. x³z³
76. x²yz³
77. xy²z³
78. y³z³
79. x²z⁴
80. xyz⁴
81. y²z⁴
82. xz⁵
83. yz⁵
84. z⁶ (end of 6th order)
85. x⁷
86. x⁶y
87. x⁵y²
88. x⁴y³
89. x³y⁴
90. x²y⁵
91. xy⁶
92. y⁷
93. x⁶z
94. x⁵yz
95. x⁴y²z
96. x³y³z
97. x²y⁴z
98. xy⁵z
99. y⁶z
100. x⁵z²
101. x⁴yz²
102. x³y²z²
103. x²y³z²
104. xy⁴z²
105. y⁵z²
106. x⁴z³
107. x³yz³
108. x²y²z³
109. xy³z³
110. y⁴z³
111. x³z⁴
112. x²yz⁴
113. xy²z⁴
114. y³z⁴
115. x²z⁵
116. xyz⁵
117. y²z⁵
118. xz⁶
119. yz⁶
120. z⁷ (end of 7th order)
121. x⁸
122. x⁷y
123. x⁶y²
124. x⁵y³
125. x⁴y⁴
126. x³y⁵
127. x²y⁶
128. xy⁷
129. y⁸
130. x⁷z
131. x⁶yz
132. x⁵y²z
133. x⁴y³z
134. x³y⁴z
135. x²y⁵z
136. xy⁶z
137. y⁷z
138. x⁶z²
139. x⁵yz²
140. x⁴y²z²
141. x³y³z²
142. x²y⁴z²
143. xy⁵z²
144. y⁶z²
145. x⁵z³
146. x⁴yz³
147. x³y²z³
148. x²y³z³
149. xy⁴z³
150. y⁵z³
151. x⁴z⁴
152. x³yz⁴
153. x²y²z⁴
154. xy³z⁴
155. y⁴z⁴
156. x³z⁵
157. x²yz⁵
158. xy²z⁵
159. y³z⁵
160. x²z⁶
161. xyz⁶
162. y²z⁶
163. xz⁷
164. yz⁷
165. z⁸ (end of 8th order)
166. x⁹
167. x⁸y
168. x⁷y²
169. x⁶y³
170. x⁵y⁴
171. x⁴y⁵
172. x³y⁶
173. x²y⁷
174. xy⁸
175. y⁹
176. x⁸z
177. x⁷yz
178. x⁶y²z
179. x⁵y³z
180. x⁴y⁴z
181. x³y⁵z
182. x²y⁶z
183. xy⁷z
184. y⁸z
185. x⁷z²
186. x⁶yz²
187. x⁵y²z²
188. x⁴y³z²
189. x³y⁴z²
190. x²y⁵z²
191. xy⁶z²
192. y⁷z²
193. x⁶z³
194. x⁵yz³
195. x⁴y²z³
196. x³y³z³
197. x²y⁴z³
198. xy⁵z³
199. y⁶z³
200. x⁵z⁴
201. x⁴yz⁴
202. x³y²z⁴
203. x²y³z⁴
204. xy⁴z⁴
205. y⁵z⁴
206. x⁴z⁵
207. x³yz⁵
208. x²y²z⁵
209. xy³z⁵
210. y⁴z⁵
211. x³z⁶
212. x²yz⁶
213. xy²z⁶
214. y³z⁶
215. x²z⁷
216. xyz⁷
217. y²z⁷
218. xz⁸
219. yz⁸
220. z⁹ (end of 9th order)
221. x¹⁰
222. x⁹y
223. x⁸y²
224. x⁷y³
225. x⁶y⁴
226. x⁵y⁵
227. x⁴y⁶
228. x³y⁷
229. x²y⁸
230. xy⁹
231. y¹⁰
232. x⁹z
233. x⁸yz
234. x⁷y²z
235. x⁶y³z
236. x⁵y⁴z
237. x⁴y⁵z
238. x³y⁶z
239. x²y⁷z
240. xy⁸z
241. y⁹z
242. x⁸z²
243. x⁷yz²
244. x⁶y²z²
245. x⁵y³z²
246. x⁴y⁴z²
247. x³y⁵z²
248. x²y⁶z²
249. xy⁷z²
250. y⁸z²
251. x⁷z³
252. x⁶yz³
253. x⁵y²z³
254. x⁴y³z³
255. x³y⁴z³
256. x²y⁵z³
257. xy⁶z³
258. y⁷z³
259. x⁶z⁴
260. x⁵yz⁴
261. x⁴y²z⁴
262. x³y³z⁴
263. x²y⁴z⁴
264. xy⁵z⁴
265. y⁶z⁴
266. x⁵z⁵
267. x⁴yz⁵
268. x³y²z⁵
269. x²y³z⁵
270. xy⁴z⁵
271. y⁵z⁵
272. x⁴z⁶
273. x³yz⁶
274. x²y²z⁶
275. xy³z⁶
276. y⁴z⁶
277. x³z⁷
278. x²yz⁷
279. xy²z⁷
280. y³z⁷
281. x²z⁸
282. xyz⁸
283. y²z⁸
284. xz⁹
285. yz⁹
286. z¹⁰ (end of 10th order)
287. x¹¹
288. x¹⁰y
289. x⁹y²
290. x⁸y³
291. x⁷y⁴
292. x⁶y⁵
293. x⁵y⁶
294. x⁴y⁷
295. x³y⁸
296. x²y⁹
297. xy¹⁰
298. y¹¹
299. x¹⁰z
300. x⁹yz
301. x⁸y²z
302. x⁷y³z
303. x⁶y⁴z
304. x⁵y⁵z
305. x⁴y⁶z
306. x³y⁷z
307. x²y⁸z
308. xy⁹z
309. y¹⁰z
310. x⁹z²
311. x⁸yz²
312. x⁷y²z²
313. x⁶y³z²
314. x⁵y⁴z²
315. x⁴y⁵z²
316. x³y⁶z²
317. x²y⁷z²
318. xy⁸z²
319. y⁹z²
320. x⁸z³
321. x⁷yz³
322. x⁶y²z³
323. x⁵y³z³
324. x⁴y⁴z³
325. x³y⁵z³
326. x²y⁶z³
327. xy⁷z³
328. y⁸z³
329. x⁷z⁴
330. x⁶yz⁴
331. x⁵y²z⁴
332. x⁴y³z⁴
333. x³y⁴z⁴
334. x²y⁵z⁴
335. xy⁶z⁴
336. y⁷z⁴
337. x⁶z⁵
338. x⁵yz⁵
339. x⁴y²z⁵
340. x³y³z⁵
341. x²y⁴z⁵
342. xy⁵z⁵
343. y⁶z⁵
344. x⁵z⁶
345. x⁴yz⁶
346. x³y²z⁶
347. x²y³z⁶
348. xy⁴z⁶
349. y⁵z⁶
350. x⁴z⁷
351. x³yz⁷
352. x²y²z⁷
353. xy³z⁷
354. y⁴z⁷
355. x³z⁸
356. x²yz⁸
357. xy²z⁸
358. y³z⁸
359. x²z⁹
360. xyz⁹
361. y²z⁹
362. xz¹⁰
363. yz¹⁰
364. z¹¹ (end of 11th order)
365. x¹²
366. x¹¹y
367. x¹⁰y²
368. x⁹y³
369. x⁸y⁴
370. x⁷y⁵
371. x⁶y⁶
372. x⁵y⁷
373. x⁴y⁸
374. x³y⁹
375. x²y¹⁰
376. xy¹¹
377. y¹²
378. x¹¹z
379. x¹⁰yz
380. x⁹y²z
381. x⁸y³z
382. x⁷y⁴z
383. x⁶y⁵z
384. x⁵y⁶z
385. x⁴y⁷z
386. x³y⁸z
387. x²y⁹z
388. xy¹⁰z
389. y¹¹z
390. x¹⁰z²
391. x⁹yz²
392. x⁸y²z²
393. x⁷y³z²
394. x⁶y⁴z²
395. x⁵y⁵z²
396. x⁴y⁶z²
397. x³y⁷z²
398. x²y⁸z²
399. xy⁹z²
400. y¹⁰z²
401. x⁹z³
402. x⁸yz³
403. x⁷y²z³
404. x⁶y³z³
405. x⁵y⁴z³
406. x⁴y⁵z³
407. x³y⁶z³
408. x²y⁷z³
409. xy⁸z³
410. y⁹z³
411. x⁸z⁴
412. x⁷yz⁴
413. x⁶y²z⁴
414. x⁵y³z⁴
415. x⁴y⁴z⁴
416. x³y⁵z⁴
417. x²y⁶z⁴
418. xy⁷z⁴
419. y⁸z⁴
420. x⁷z⁵
421. x⁶yz⁵
422. x⁵y²z⁵
423. x⁴y³z⁵
424. x³y⁴z⁵
425. x²y⁵z⁵
426. xy⁶z⁵
427. y⁷z⁵
428. x⁶z⁶
429. x⁵yz⁶
430. x⁴y²z⁶
431. x³y³z⁶
432. x²y⁴z⁶
433. xy⁵z⁶
434. y⁶z⁶
435. x⁵z⁷
436. x⁴yz⁷
437. x³y²z⁷
438. x²y³z⁷
439. xy⁴z⁷
440. y⁵z⁷
441. x⁴z⁸
442. x³yz⁸
443. x²y²z⁸
444. xy³z⁸
445. y⁴z⁸
446. x³z⁹
447. x²yz⁹
448. xy²z⁹
449. y³z⁹
450. x²z¹⁰
451. xyz¹⁰
452. y²z¹⁰
453. xz¹¹
454. yz¹¹
455. z¹² (end of 12th order)

Representation in initialization files

AIR no longer uses ASCII files to initialize polynomial warps. Instead, a .warp file can be used to initialize polynomials having an order one greater than the order of the .warp file

Default initialization

The default initialization for this model was modified in AIR 5.1 to make it identical to the default for alignlinear even when voxel sizes in the two files are different

If no initialization .warp file is specified, the default initialization is:

• k_x1 =((rx_dim-1) - (sx_dim-1)*(sx_size / rx_size)) / 2
• k_x2 =sx_size / rx_size
• k_y1 =((ry_dim-1) - (sy_dim-1)*(sy_size / ry_size)) / 2
• k_y2 =sy_size / ry_size
• k_z1 =((rz_dim-1) - (sz_dim-1)*(sz_size / rz_size)) / 2
• k_z2 =sz_size / rz_size
• All other parameters=0

where:

• sx_size is the voxel x size of the standard file
• sy_size is the voxel y size of the standard file
• sz_size is the voxel z size of the standard file
• rx_size is the voxel x size of the reslice file
• ry_size is the voxel y size of the reslice file
• rz_size is the voxel z size of the reslice file
• sx_dim is the x dimension of the standard file
• sy_dim is the y dimension of the standard file
• sz_dim is the z dimension of the standard file
• rx_dim is the x dimension of the reslice file
• ry_dim is the y dimension of the reslice file
• rz_dim is the z dimension of the reslice file

This results in the exact centers of the two files being aligned to one another.